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Applying the principle of homogeneity of dimensions, determine which one is correct, where $$T$$ is time period, $$G$$ is gravitational constant, $$M$$ is mass, $$r$$ is radius of orbit.
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A cyclist starts from the point $$P$$ of a circular ground of radius $$2$$ km and travels along its circumference to the point $$S$$. The displacement of a cyclist is:
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A $$2$$ kg brick begins to slide over a surface which is inclined at an angle of $$45°$$ with respect to horizontal axis. The co-efficient of static friction between their surfaces is:
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A body of $$m$$ kg slides from rest along the curve of vertical circle from point $$A$$ to $$B$$ in friction less path. The velocity of the body at $$B$$ is: (given, $$R = 14$$ m, $$g = 10$$ m/s$$^2$$ and $$\sqrt{2} = 1.4$$)
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A $$90$$ kg body placed at $$2R$$ distance from surface of earth experiences gravitational pull of: ($$R$$ = Radius of earth, $$g = 10$$ m s$$^{-2}$$)
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Correct formula for height of a satellite from earth's surface is:
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Given below are two statements: Statement I: The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well. Statement II: The rise of a liquid in a capillary tube does not depend on the inner radius of the tube. In the light of the above statements, choose the correct answer from the options given below:
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A sample of gas at temperature $$T$$ is adiabatically expanded to double its volume. Adiabatic constant for the gas is $$\gamma = 3/2$$. The work done by the gas in the process is: ($$\mu = 1$$ mole)
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The translational degrees of freedom $$(f_t)$$ and rotational degrees of freedom $$(f_r)$$ of $$CH_4$$ molecule are:
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In simple harmonic motion, the total mechanical energy of given system is $$E$$. If mass of oscillating particle $$P$$ is doubled then the new energy of the system for same amplitude is
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A charge $$q$$ is placed at the center of one of the surface of a cube. The flux linked with the cube is:
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An electric bulb rated $$50$$ W $$- 200$$ V is connected across a $$100$$ V supply. The power dissipation of the bulb is:
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The magnetic moment of a bar magnet is $$0.5$$ Am$$^2$$. It is suspended in a uniform magnetic field of $$8 \times 10^{-2}$$ T. The work done in rotating it from its most stable to most unstable position is:
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Match List I with List II:
Choose the correct answer from the options given below:
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Arrange the following in the ascending order of wavelength: A. Gamma rays $$(\lambda_1)$$, B. x-rays $$(\lambda_2)$$, C. Infrared waves $$(\lambda_3)$$, D. Microwaves $$(\lambda_4)$$. Choose the most appropriate answer from the options given below:
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The width of one of the two slits in a Young's double slit experiment is 4 times that of the other slit. The ratio of the maximum of the minimum intensity in the interference pattern is:
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Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Number of photons increases with increase in frequency of light. Reason R: Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation. In the light of the above statements, choose the most appropriate answer from the options given below:
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According to Bohr's theory, the moment of momentum of an electron revolving in $$4^{th}$$ orbit of hydrogen atom is:
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Identify the logic gate given in the circuit:
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Which of the diode circuit shows correct biasing used for the measurement of dynamic resistance of p-n junction diode:
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A bus moving along a straight highway with speed of $$72$$ km/h is brought to halt within $$4$$ s after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is _____ m.
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In a system two particles of masses $$m_1 = 3$$ kg and $$m_2 = 2$$ kg are placed at certain distance from each other. The particle of mass $$m_1$$ is moved towards the center of mass of the system through a distance $$2$$ cm. In order to keep the center of mass of the system at the original position, the particle of mass $$m_2$$ should move towards the center of mass by the distance _____ cm.
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Mercury is filled in a tube of radius $$2$$ cm up to a height of $$30$$ cm. The force exerted by mercury on the bottom of the tube is _____ N. (Given, atmospheric pressure $$= 10^5$$ Nm$$^{-2}$$, density of mercury $$= 1.36 \times 10^4$$ kg m$$^{-3}$$, $$g = 10$$ m s$$^{-2}$$, $$\pi = \frac{22}{7}$$)
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The displacement of a particle executing SHM is given by $$x = 10 \sin\left(\omega t + \frac{\pi}{3}\right)$$ m. The time period of motion is $$3.14$$ s. The velocity of the particle at $$t = 0$$ is _____ m/s.
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A parallel plate capacitor of capacitance $$12.5$$ pF is charged by a battery connected between its plates to potential difference of $$12.0$$ V. The battery is now disconnected and a dielectric slab ($$\epsilon_r = 6$$) is inserted between the plates. The change in its potential energy after inserting the dielectric slab is _____ $$ 10^{-12}$$ J.
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Two wires $$A$$ and $$B$$ are made up of the same material and have the same mass. Wire $$A$$ has radius of $$2.0$$ mm and wire $$B$$ has radius of $$4.0$$ mm. The resistance of wire $$B$$ is $$2\Omega$$. The resistance of wire $$A$$ is _____ $$\Omega$$.
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Two parallel long current carrying wire separated by a distance $$2r$$ are shown in the figure. The ratio of magnetic field at $$A$$ to the magnetic field produced at $$C$$ is $$\frac{x}{7}$$. The value of $$x$$ is _____
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A rod of length $$60$$ cm rotates with a uniform angular velocity $$20$$ rad s$$^{-1}$$ about its perpendicular bisector, in a uniform magnetic field $$0.5$$ T. The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is _____ V.
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A light ray is incident on a glass slab of thickness $$4\sqrt{3}$$ cm and refractive index $$\sqrt{2}$$. The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is _____ cm. (Given $$\sin 15° = 0.25$$)
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The disintegration energy $$Q$$ for the nuclear fission of $$^{235}U \rightarrow ^{140}Ce + ^{94}Zr + n$$ is _____ MeV. Given atomic masses of $$^{235}U : 235.0439$$ u; $$^{140}Ce : 139.9054$$ u; $$^{94}Zr : 93.9063$$ u; $$n : 1.0086$$ u, Value of $$c^2 = 931$$ MeV/u
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Choose the Incorrect Statement about Dalton's Atomic Theory
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The correct order of the first ionization enthalpy is
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Given below are two statements: Statement I: The correct order of first ionization enthalpy values of Li, Na, F and Cl is $$Na < Li < Cl < F$$. Statement II: The correct order of negative electron gain enthalpy values of Li, Na, F and Cl is $$Na < Li < F < Cl$$. In the light of the above statements, choose the correct answer from the options given below:
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The correct statement/s about Hydrogen bonding is/are: A. Hydrogen bonding exists when H is covalently bonded to the highly electronegative atom. B. Intermolecular H bonding is present in o-nitrophenol. C. Intramolecular H bonding is present in HF. D. The magnitude of H bonding depends on the physical state of the compound. E. H-bonding has powerful effect on the structure and properties of compounds. Choose the correct answer from the options given below:
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The number of species from the following that have pyramidal geometry around the central atom is _______. $$S_2O_3^{2-}$$, $$SO_4^{2-}$$, $$SO_3^{2-}$$, $$S_2O_7^{2-}$$
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The equilibrium constant for the reaction $$SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2}O_2(g)$$ is $$K_c = 4.9 \times 10^{-2}$$. The value of $$K_c$$ for the reaction given below is $$2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$$:
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Correct order of stability of carbanion is
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Common name of Benzene-1,2-diol is:
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The adsorbent used in adsorption chromatography is/are: A. silica gel, B. alumina, C. quick lime, D. magnesia. Choose the most appropriate answer from the options given below:
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In the above chemical reaction sequence "A" and "B" respectively are:
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For a strong electrolyte, a plot of molar conductivity against (concentration)$$^{1/2}$$ is a straight line, with a negative slope, the correct unit for the slope is:
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Fuel cell, using hydrogen and oxygen as fuels: A. has been used in spaceship, B. has an efficiency of 40% to produce electricity, C. uses aluminum as catalysts, D. is eco-friendly, E. is actually a type of Galvanic cell only. Choose the correct answer from the options given below:
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When $$MnO_2$$ and $$H_2SO_4$$ is added to a salt (A), the greenish yellow gas liberated as salt (A) is:
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A first row transition metal in its +2 oxidation state has a spin-only magnetic moment value of $$3.86$$ BM. The atomic number of the metal is:
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If an iron (III) complex with the formula $$[Fe(NH_3)_x(CN)_y]^-$$ has no electron in its $$e_g$$ orbital, then the value of $$x + y$$ is:
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The number of unpaired d-electrons in $$[Co(H_2O)_6]^{3+}$$ is:
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$$CH_3-CH_2-CH_2-Br + NaOH \xrightarrow{C_2H_5OH}$$ Product 'A'.
Consider the above reactions, identify product $$B$$ and product $$C$$.
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Find out the major product formed from the following reaction. [Me: $$-CH_3$$]
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Match List I with List II:
Choose the correct answer from the options given below:
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The maximum number of orbitals which can be identified with $$n = 4$$ and $$m_l = 0$$ is ______
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Number of compounds/species from the following with non-zero dipole moment is ______. $$BeCl_2$$, $$BCl_3$$, $$NF_3$$, $$XeF_4$$, $$CCl_4$$, $$H_2O$$, $$H_2S$$, $$HBr$$, $$CO_2$$, $$H_2$$, $$HCl$$
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Three moles of an ideal gas are compressed isothermally from $$60$$ L to $$20$$ L using constant pressure of $$5$$ atm. Heat exchange $$Q$$ for the compression is $$-$$ ______ Lit. atm.
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The total number of 'sigma' and 'Pi' bonds in 2-oxohex-4-ynoic acid is ______
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$$2.7$$ kg of each of water and acetic acid are mixed. The freezing point of the solution will be $$-x°C$$. Consider the acetic acid does not dimerise in water, nor dissociates in water. $$x =$$ ______ (nearest integer) [Given: Molar mass of water $$= 18$$ g mol$$^{-1}$$, acetic acid $$= 60$$ g mol$$^{-1}$$, $$K_f$$ H$$_2$$O: $$1.86$$ K kg mol$$^{-1}$$, $$K_f$$ acetic acid: $$3.90$$ K kg mol$$^{-1}$$, freezing point: H$$_2$$O $$= 273$$ K, acetic acid $$= 290$$ K]
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Consider the following reaction, the rate expression of which is given below:
$$A + B \rightarrow C$$, rate$$ = k[A]^{1/2}[B]^{1/2}$$.
The reaction is initiated by taking concentration of $$1$$M of $$A$$ and $$B$$ each. If the rate constant $$(k)$$ is $$4.6 \times 10^{-2}$$ s$$^{-1}$$, then the time taken for $$A$$ to become $$0.1$$M is ______ sec. (nearest integer)
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A first row transition metal with highest enthalpy of atomisation, upon reaction with oxygen at high temperature forms oxides of formula $$M_2O_n$$ (where $$n = 3, 4, 5$$). The 'spin-only' magnetic moment value of the amphoteric oxide from the above oxides is ______ BM (nearest integer). (Given atomic number: Sc: 21, Ti: 22, V: 23, Cr: 24, Mn: 25, Fe: 26, Co: 27, Ni: 28, Cu: 29, Zn: 30)
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Phthalimide is made to undergo following sequence of reactions:
Total number of $$\pi$$ bonds present in product 'P' is/are ______
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From $$6.55$$ g of aniline, the maximum amount of acetanilide that can be prepared will be ______ $$\times 10^{-1}$$ g.
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Vanillin compound obtained from vanilla beans, has total sum of oxygen atoms and $$\pi$$ electrons is ______
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The area (in sq. units) of the region $$S = \{z \in \mathbb{C} : |z - 1| \leq 2; (z + \bar{z}) + i(z - \bar{z}) \leq 2, \text{Im}(z) \geq 0\}$$ is
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The value of $$\frac{1 \times 2^2 + 2 \times 3^2 + \ldots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \ldots + 100^2 \times 101}$$ is
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Let three real numbers $$a, b, c$$ be in arithmetic progression and $$a + 1, b, c + 3$$ be in geometric progression. If $$a > 10$$ and the arithmetic mean of $$a, b$$ and $$c$$ is $$8$$, then the cube of the geometric mean of $$a, b$$ and $$c$$ is
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If the coefficients of $$x^4$$, $$x^5$$ and $$x^6$$ in the expansion of $$(1 + x)^n$$ are in the arithmetic progression, then the maximum value of $$n$$ is:
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Let $$C$$ be a circle with radius $$\sqrt{10}$$ units and centre at the origin. Let the line $$x + y = 2$$ intersects the circle $$C$$ at the points $$P$$ and $$Q$$. Let $$MN$$ be a chord of $$C$$ of length $$2$$ unit and slope $$-1$$. Then, a distance (in units) between the chord $$PQ$$ and the chord $$MN$$ is
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Let $$PQ$$ be a chord of the parabola $$y^2 = 12x$$ and the midpoint of $$PQ$$ be at $$(4, 1)$$. Then, which of the following point lies on the line passing through the points $$P$$ and $$Q$$?
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Consider a hyperbola $$H$$ having centre at the origin and foci on the $$x$$-axis. Let $$C_1$$ be the circle touching the hyperbola $$H$$ and having the centre at the origin. Let $$C_2$$ be the circle touching the hyperbola $$H$$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $$C_1$$ and $$C_2$$ are $$36\pi$$ and $$4\pi$$, respectively, then the length (in units) of latus rectum of $$H$$ is
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Let $$f(x) = \int_0^x (t + \sin(1 - e^t))dt$$, $$x \in \mathbb{R}$$. Then, $$\lim_{x \to 0} \frac{f(x)}{x^3}$$ is equal to
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If the mean of the following probability distribution of a random variable $$X$$:
is $$\frac{46}{9}$$, then the variance of the distribution is
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Let a relation $$R$$ on $$\mathbb{N} \times \mathbb{N}$$ be defined as: $$(x_1, y_1) R (x_2, y_2)$$ if and only if $$x_1 \leq x_2$$ or $$y_1 \leq y_2$$. Consider the two statements: (I) $$R$$ is reflexive but not symmetric. (II) $$R$$ is transitive. Then which one of the following is true?
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Let $$A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$$ and $$B = I + \text{adj}(A) + (\text{adj } A)^2 + \ldots + (\text{adj } A)^{10}$$. Then, the sum of all the elements of the matrix $$B$$ is:
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Given that the inverse trigonometric function assumes principal values only. Let $$x, y$$ be any two real numbers in $$[-1, 1]$$ such that $$\cos^{-1} x - \sin^{-1} y = \alpha$$, $$\frac{-\pi}{2} \leq \alpha \leq \pi$$. Then, the minimum value of $$x^2 + y^2 + 2xy \sin \alpha$$ is
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If the function $$f(x) = \begin{cases} \frac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_e 2 \log_e 3, & x = 0 \end{cases}$$ is continuous at $$x = 0$$, then the value of $$a^2$$ is equal to
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Let $$f(x) = 3\sqrt{x - 2} + \sqrt{4 - x}$$ be a real valued function. If $$\alpha$$ and $$\beta$$ are respectively the minimum and the maximum values of $$f$$, then $$\alpha^2 + 2\beta^2$$ is equal to
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If the value of the integral $$\int_{-1}^{1} \frac{\cos \alpha x}{1 + 3^x} dx$$ is $$\frac{2}{\pi}$$. Then, a value of $$\alpha$$ is
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The area (in sq. units) of the region described by $$\{(x, y) : y^2 \leq 2x, y \geq 4x - 1\}$$ is
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Let $$y = y(x)$$ be the solution of the differential equation $$(x^2 + 4)^2 dy + (2x^3 y + 8xy - 2)dx = 0$$. If $$y(0) = 0$$, then $$y(2)$$ is equal to
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Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k}$$ and $$\vec{c} = x\hat{i} + 2\hat{j} + 3\hat{k}$$, $$x \in \mathbb{R}$$. If $$\vec{d}$$ is the unit vector in the direction of $$\vec{b} + \vec{c}$$ such that $$\vec{a} \cdot \vec{d} = 1$$, then $$(\vec{a} \times \vec{b}) \cdot \vec{c}$$ is equal to
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For $$\lambda > 0$$, let $$\theta$$ be the angle between the vectors $$\vec{a} = \hat{i} + \lambda\hat{j} - 3\hat{k}$$ and $$\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$$. If the vectors $$\vec{a} + \vec{b}$$ and $$\vec{a} - \vec{b}$$ are mutually perpendicular, then the value of $$(14 \cos \theta)^2$$ is equal to
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Let $$P$$ be the point of intersection of the lines $$\frac{x-2}{1} = \frac{y-4}{5} = \frac{z-2}{1}$$ and $$\frac{x-3}{2} = \frac{y-2}{3} = \frac{z-3}{2}$$. Then, the shortest distance of $$P$$ from the line $$4x = 2y = z$$ is
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There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____
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Let $$S = \{\sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0$$ has real roots$$\}$$. If $$\alpha$$ and $$\beta$$ be the smallest and largest elements of the set $$S$$, respectively, then $$3((\alpha - 2)^2 + (\beta - 1)^2)$$ equals _____
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Consider a triangle $$ABC$$ having the vertices $$A(1, 2)$$, $$B(\alpha, \beta)$$ and $$C(\gamma, \delta)$$ and angles $$\angle ABC = \frac{\pi}{6}$$ and $$\angle BAC = \frac{2\pi}{3}$$. If the points $$B$$ and $$C$$ lie on the line $$y = x + 4$$, then $$\alpha^2 + \gamma^2$$ is equal to _____
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Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix}$$ and the determinant of $$A$$ be $$1$$. If $$A^{-1} = \alpha A + \beta I$$, where $$I$$ is an identity matrix of order $$2 \times 2$$, then $$\alpha + \beta$$ equals _____
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Consider the function $$f : \mathbb{R} \to \mathbb{R}$$ defined by $$f(x) = \frac{2x}{\sqrt{1 + 9x^2}}$$. If the composition of $$f$$, $$\underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text{ times}}(x) = \frac{2^{10}x}{\sqrt{1 + 9\alpha x^2}}$$, then the value of $$\sqrt{3\alpha + 1}$$ is equal to _____
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a thrice differentiable function such that $$f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2$$ and $$f(4) = -2$$. Then, the minimum number of zeros of $$(3f'f'' + ff''')(x)$$ is _____
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If $$\int \csc^5 x \, dx = \alpha \cot x \csc x \left(\csc^2 x + \frac{3}{2}\right) + \beta \log_e \left|\tan \frac{x}{2}\right| + C$$ where $$\alpha, \beta \in \mathbb{R}$$ and $$C$$ is the constant of integration, then the value of $$8(\alpha + \beta)$$ equals _____
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Let $$y = y(x)$$ be the solution of the differential equation $$(x + y + 2)^2 dx = dy$$, $$y(0) = -2$$. Let the maximum and minimum values of the function $$y = y(x)$$ in $$\left[0, \frac{\pi}{3}\right]$$ be $$\alpha$$ and $$\beta$$, respectively. If $$(3\alpha + \pi)^2 + \beta^2 = \gamma + \delta\sqrt{3}$$, $$\gamma, \delta \in \mathbb{Z}$$, then $$\gamma + \delta$$ equals _____
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Consider a line $$L$$ passing through the points $$P(1, 2, 1)$$ and $$Q(2, 1, -1)$$. If the mirror image of the point $$A(2, 2, 2)$$ in the line $$L$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + 6\gamma$$ is equal to _____
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In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $$\frac{1}{3}$$ and $$\frac{2}{3}$$ respectively. Let $$x$$ be the number of matches that the team wins, and $$y$$ be the number of matches that team loses. If the probability $$P(|x - y| \leq 2)$$ is $$p$$, then $$3^9 p$$ equals _____
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